Eberlein almost periodic solutions for some evolution equations with monotonicity

This paper investigates the existence of Eberlein weakly almost periodic solutions for differential equations of the form \(u'=Au+f(t)\) and \(u'=A(t)u+f(t)\). In the first scenario, when A generates a strongly asymptotically semigroup, we establish the existence of Eberlein-weakly almost periodic solutions, thereby extending and improving a previous result in Zaidman(Ann Univ Ferrara 14(1): 29–34, 1969). In the second case, we consider a more general situation where A(t) is a (possibly nonlinear) operator satisfying a monotony condition. Unlike most existing works in the literature, our approach does not rely on tools of exponential dichotomy and Lipschitz nonlinearity. Lastly, we illustrate the practical relevance of our findings by presenting real-world models, including a hematopoiesis model, that exemplify the key findings. A numerical simulation is also provided.